Finite Group Databasebeta

← #211901

Subgroups — #211901

Order 512.

581 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22166C211yesmaximal, characteristic
2#22131C211yesmaximal, characteristic, complemented
3#19156C211yesmaximal, characteristic
4#17957C211yesmaximal, characteristic, complemented
5#17328C211yesmaximal, characteristic, complemented
6#9722C211yesmaximal, characteristic, complemented
7#9625C211yesmaximal, characteristic, complemented
8#9590C211yesmaximal, characteristic, complemented
9#9589C211yesmaximal, characteristic, complemented
10#9208C211yesmaximal, characteristic
11#9207C211yesmaximal, characteristic, complemented
12#8668C211yesmaximal, characteristic, complemented
13#8282C211yesmaximal, characteristic, complemented
14#8254C211yesmaximal, characteristic, complemented
15#8253C211yesmaximal, characteristic, complemented
16#3167C2211yescharacteristic
17#3076C2211yescharacteristic
18#3035C2211yescharacteristic
19#2965C2211yescharacteristic
20C8×#41C2211yescharacteristic
21#2960C2211yescharacteristic
22#2931C2211yescharacteristic
23#2922C2211yescharacteristic
24#1986C2211yescharacteristic
25#1980C2211yescharacteristic
26#1975—21no—
27#1950C2211yescharacteristic
28#1950C2211yescharacteristic
29#1946—21no—
30#1938—21no—
31#1938—21no—
32#1929—21no—
33#1922—21no—
34#1876—21no—
35#1866—21no—
36#1806C2211yescharacteristic
37#1806C2211yescharacteristic
38C4×#107C2211yescharacteristic
39#1770C2211yescharacteristic
40#1770C2211yescharacteristic
41#1700C411yescharacteristic, complemented
42#1699C411yescharacteristic, complemented
43#1698C411yescharacteristic, complemented
44#1697C411yescharacteristic, complemented
45#1692C411yescharacteristic, complemented
46#1691C411yescharacteristic, complemented
47#1686C411yescharacteristic, complemented
48#1685C411yescharacteristic, complemented
49#1590—21no—
50#1590—21no—
51C8×SD16—21no—
52C8×SD16—21no—
53#1566C2211yescharacteristic
54C2×#335C2211yescharacteristic
55#1542C2211yescharacteristic
56#1540C2211yescharacteristic
57#1503C2211yescharacteristic
58#1502C2211yescharacteristic
59#1493C2211yescharacteristic
60#1492C2211yescharacteristic
61#1454—21no—
62#1453—21no—
63#1452C2211yescharacteristic
64#1451C2211yescharacteristic
65#1450—21no—
66C8×M16—21no—
67C8×M16C2211yescharacteristic
68C2×C82C2211yescharacteristic
69#1303C2211yescharacteristic
70#1303C2211yescharacteristic
71#1301C2211yescharacteristic
72#1301C2211yescharacteristic
73#1279C2211yescharacteristic
74#1279C2211yescharacteristic
75C2×#135—21no—
76C2×#131—21no—
77#533C2311yescharacteristic
78#532C2311yescharacteristic
79C4×#41C2311yescharacteristic
80#498C2×C411yescharacteristic
81#492C2×C411yescharacteristic
82#487C2×C411yescharacteristic
83#486C2×C411yescharacteristic
84#479—21no—
85#478—21no—
86#477C2×C411yescharacteristic
87#476C2×C411yescharacteristic
88#475C2×C411yescharacteristic
89#474C2×C411yescharacteristic
90#465—21no—
91#465—21no—
92#464—21no—
93#463—21no—
94#462C2×C411yescharacteristic
95#461C2×C411yescharacteristic
96#460C2×C411yescharacteristic
97#459C2×C411yescharacteristic
98#446—21no—
99#446—21no—
100#446C2×C411yescharacteristic
101#446C2×C411yescharacteristic
102C8×Q8—21no—
103C8×Q8—21no—
104C8×Q8—21no—
105C4×SD16—21no—
106C4×SD16—21no—
107#436—21no—
108#436—21no—
109#436C2×C411yescharacteristic
110#436C2×C411yescharacteristic
111#435—21no—
112#435—21no—
113C8×D4—21no—
114C8×D4—22no—
115C8×D4—21no—
116C8×D4—21no—
117#433—21no—
118#433—21no—
119#431—21no—
120#431—21no—
121#431—21no—
122#431—21no—
123#431C2311yescharacteristic
124#431C2311yescharacteristic
125C2×#108—21no—
126C2×#108—21no—
127#427—21no—
128C2×#107—21no—
129#424—21no—
130#424D411yescharacteristic
131#424D411yescharacteristic
132#423C2311yescharacteristic
133#423C2311yescharacteristic
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2311yescharacteristic
136C2×#104—21no—
137C2×#104—21no—
138C2×#104—21no—
139C2×#104—21no—
140#416—21no—
141#416—21no—
142#408—21no—
143#408D411yescharacteristic
144#408D411yescharacteristic
145#405—21no—
146C4×M16—21no—
147C4×M16C2311yescharacteristic
148C2×#97—21no—
149C2×#97—21no—
150C2×C4×C8—21no—
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C4×C4⋊C4C2311yescharacteristic
155C4×C4⋊C4C2311yescharacteristic
156#341—21no—
157#341—21no—
158#340—21no—
159#339—21no—
160#339—21no—
161#339—21no—
162#339—21no—
163#336—21no—
164#336—21no—
165#335C2×C411yescharacteristic
166#335C2×C411yescharacteristic
167#335C2×C411yescharacteristic
168#335C2×C411yescharacteristic
169#334—21no—
170#334—21no—
171#333C2×C411yescharacteristic
172#333C2×C411yescharacteristic
173#332C2×C411yescharacteristic
174#332C2×C411yescharacteristic
175#332C2×C411yescharacteristic
176#332C2×C411yescharacteristic
177#331C2×C411yescharacteristic
178#331C2×C411yescharacteristic
179#328—22no—
180#327—22no—
181Q8⋊C8—21no—
182Q8⋊C8—21no—
183D4⋊C8—21no—
184D4⋊C8—21no—
185#322—21no—
186#322—21no—
187#322—21no—
188#322—21no—
189#322—21no—
190C82—21no—
191C82D411yescharacteristic
192C82D411yescharacteristic
193C82D411yescharacteristic
194C82D411yescharacteristic
195C2×#41—21no—
196C2×#41C2×D411yescharacteristic
197#135—81no—
198#135—81no—
199C2×Q16—21no—
200C2×SD16—21no—
201C2×SD16—21no—
202C2×SD16—21no—
203C2×SD16—21no—
204C2×D8—21no—
205#131—81no—
206#131—81no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—41no—
212C2×M16C2×D411yescharacteristic
213C2×M16C2×D411yescharacteristic
214C22×C8—22no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—41no—
219C22×C8C2×D411yescharacteristic
220C22×C8C2×D411yescharacteristic
221C22×C8#4111yescharacteristic
222C22×C8#4111yescharacteristic
223C4⋊Q8C22×C411yescharacteristic
224C4⋊Q8C22×C411yescharacteristic
225C4⋊Q8C22×C411yescharacteristic
226#127C22×C411yescharacteristic
227#125—21no—
228#124—21no—
229C22⋊Q8—21no—
230C22⋊Q8—21no—
231#121—21no—
232#121—21no—
233C4×Q8—21no—
234C4×Q8—21no—
235C4×Q8C22×C411yescharacteristic
236C4×Q8#4111yescharacteristic
237C4×D4—21no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—21no—
241C4×D4C22×C411yescharacteristic
242C4×D4#4111yescharacteristic
243#117—21no—
244#117—21no—
245#117C2×D411yescharacteristic
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42C2411yescharacteristic
252#108—81no—
253#108—81no—
254#107—21no—
255#107—21no—
256#107—42no—
257#106—21no—
258#106—21no—
259#106—21no—
260#106—21no—
261C4⋊C8—22no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—41no—
276C4⋊C8—41no—
277C4⋊C8C22×C411yescharacteristic
278C4⋊C8C22×C411yescharacteristic
279C4⋊C8C22×C411yescharacteristic
280C4⋊C8C22×C411yescharacteristic
281C4⋊C8C22×C411yescharacteristic
282C4⋊C8C22×C411yescharacteristic
283#104—81no—
284#104—81no—
285#104—81no—
286#104—81no—
287#103—21no—
288#103—21no—
289#103—21no—
290#103—21no—
291#103—21no—
292#103—21no—
293#102—21no—
294#102—21no—
295#102—21no—
296#102—21no—
297#102—21no—
298#102—21no—
299C22⋊C8—22no—
300C22⋊C8—21no—
301C22⋊C8—21no—
302C22⋊C8—21no—
303C22⋊C8—21no—
304C22⋊C8—21no—
305C22⋊C8—21no—
306C22⋊C8—21no—
307C22⋊C8—41no—
308C22⋊C8—41no—
309C22⋊C8#4112yes—
310#97—21no—
311#97—21no—
312#97—21no—
313#97—41no—
314#97—81no—
315#97—81no—
316C4×C8—22no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—21no—
321C4×C8—41no—
322C4×C8C22×C411yescharacteristic
323C4×C8C22×C411yescharacteristic
324C4×C8C2×D411yescharacteristic
325C4×C8C2×D411yescharacteristic
326C4×C8C2×D411yescharacteristic
327C4×C8C2×D411yescharacteristic
328C4×C8C2×D411yescharacteristic
329C4×C8C2×D411yescharacteristic
330C4×C8#4111yescharacteristic
331C4×C8#4111yescharacteristic
332#95—21no—
333#95—21no—
334#41—21no—
335#41—21no—
336#41—21no—
337#41—21no—
338#41—41no—
339#41—41no—
340#41—41no—
341#41—41no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8C4×D411yescharacteristic
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—41no—
351C2×D4C4×D411yescharacteristic
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4C22×D411yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361Q16—81no—
362SD16—42no—
363SD16—42no—
364SD16—81no—
365SD16—81no—
366D8—81no—
367M16—21no—
368M16—21no—
369M16—21no—
370M16—21no—
371M16—42no—
372M16—42no—
373M16—41no—
374M16—41no—
375M16—41no—
376M16—41no—
377M16—82no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—22no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—22no—
393C2×C8—21no—
394C2×C8—41no—
395C2×C8—42no—
396C2×C8—44no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—82no—
404C2×C8C4×D411yescharacteristic
405C2×C8C4×D411yescharacteristic
406C2×C8C4×D411yescharacteristic
407C2×C8C4×D411yescharacteristic
408C2×C8#12111yescharacteristic
409C2×C8#12111yescharacteristic
410C2×C8#12111yescharacteristic
411C2×C8#12111yescharacteristic
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4C4×D411yescharacteristic
427C4⋊C4C4×D411yescharacteristic
428C22⋊C4—21no—
429C22⋊C4—21no—
430C22⋊C4—21no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42—41no—
438C42—41no—
439C42—41no—
440C42—41no—
441C42—41no—
442C42C23×C411yescharacteristic
443C42C22×D411yescharacteristic
444C42C2×#4111yescharacteristic
445C42C2×#4111yescharacteristic
446C23—21no—
447C23—41no—
448C23#53211yescharacteristic
449Q8—21no—
450Q8—21no—
451Q8—41no—
452Q8—41no—
453Q8—41no—
454Q8—41no—
455Q8—81no—
456D4—21no—
457D4—21no—
458D4—41no—
459D4—41no—
460D4—41no—
461D4—41no—
462D4—41no—
463D4—41no—
464D4—81no—
465D4—81no—
466D4—81no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4C2×C4×D411yescharacteristic
490C2×C4C2×C4×D411yescharacteristic
491C2×C4C4×#4111yescharacteristic
492C2×C4C2×#12111yescharacteristic
493C2×C4#52511yescharacteristic
494C2×C4#52911yescharacteristic
495C2×C4#57611yescharacteristic
496C2×C4#57711yescharacteristic
497C2×C4#57711yescharacteristic
498C2×C4#57811yescharacteristic
499C8—22no—
500C8—21no—
501C8—21no—
502C8—21no—
503C8—21no—
504C8—21no—
505C8—21no—
506C8—21no—
507C8—42no—
508C8—41no—
509C8—41no—
510C8—41no—
511C8—41no—
512C8—41no—
513C8—41no—
514C8—41no—
515C8—41no—
516C8—41no—
517C8—81no—
518C8—81no—
519C8#44412yes—
520C22—21no—
521C22—21no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22—81no—
526C22—81no—
527C22—81no—
528C22C4×#12111yescharacteristic
529C22#317811yescharacteristic
530C22#317911yescharacteristic
531C4—21no—
532C4—21no—
533C4—21no—
534C4—41no—
535C4—41no—
536C4—41no—
537C4—81no—
538C4—81no—
539C4—81no—
540C4—161no—
541C4—161no—
542C4#294611yescharacteristic
543C4#294711yescharacteristic
544C4C2×#44411yescharacteristic
545C4#295511yescharacteristic
546C4#305711yescharacteristic
547C4#306111yescharacteristic
548C2—21no—
549C2—21no—
550C2—81no—
551C2—161no—
552C2#2151811yescharacteristic
553C2#2153511yescharacteristic
554C2#2155811yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C4×#121
prime radical#3035C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #22131C2semidirect
4. #17957C2semidirect
5. #17328C2semidirect
6. #9722C2semidirect
7. #9625C2semidirect
8. #9590C2semidirect
9. #9589C2semidirect
11. #9207C2semidirect
12. #8668C2semidirect
13. #8282C2semidirect
14. #8254C2semidirect
15. #8253C2semidirect
41. #1700C4semidirect
42. #1699C4semidirect
43. #1698C4semidirect
44. #1697C4semidirect
45. #1692C4semidirect
46. #1691C4semidirect
47. #1686C4semidirect
48. #1685C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#221662
#221312
#191562
#179572
#173282
#97222
#96252
#95902
#95892
#92082
#92072
#86682
#82822
#82542
#82532